We establish the smallest upper bound for the p absolute central moment over the class of all random variables with values in a compact interval. Numerical values of the bound are calculated for the ﬁrst ten integer values of p, and its asymptotic behaviour derived when p tends to inﬁnity. In addition, we establish an analogous bound in the case of all symmetric random variables with values in a compact interval. Such results play a role in a number of areas including actuarial science, economics, ﬁnance, operations research, and reliability.